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      <a id="GENfarey"></a>
      <div class="titlepage"></div>
      <a id="IndexGENfarey" class="indexterm"></a>
      <div class="refnamediv">
        <h2>
          <span class="refentrytitle">GENfarey</span>
        </h2>
        <p>"farey" — 
      Fills a table with the Farey Sequence F<sub>n</sub> of the integer n.
    </p>
      </div>
      <div class="refsect1">
        <a id="idp216250496"></a>
        <h2>Description</h2>
        <p>
      A Farey Sequence F<sub>n</sub> of order n is a list of fractions in their lowest terms between 0 and 1 and in ascending order. Their denominators do not exceed n. 
      This means a fraction a/b belongs to F<sub>n</sub> if 0 ≤ a ≤ b ≤ n.
      The numerator and denominator of each fraction are always coprime.
      0 and 1 are included in F<sub>n</sub> as the fractions 0/1 and 1/1.
      For example F<sub>5</sub> = {0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1}
      Some properties of the Farey Sequence:
      </p>
        <div class="itemizedlist">
          <ul class="itemizedlist" style="list-style-type: disc; ">
            <li class="listitem">
          If a/b and c/d are two successive terms of F<sub>n</sub>, then bc - ad = 1.
        </li>
            <li class="listitem">
          If a/b, c/d, e/f are three successive terms of F<sub>n</sub>, then:
          c/d = (a+e) / (b+f).
          In this case c/d is called the mediant fraction between a/b and e/f.
        </li>
            <li class="listitem">
          If n &gt; 1, then no two successive terms ofF<sub>n</sub> have the same denominator.
        </li>
          </ul>
        </div>
        <p>
    </p>
        <p>
      The length of any Farey Sequence F<sub>n</sub> is determined by
      |F<sub>n</sub>| = 1 + SUM over n (phi(m))
      where phi(m) is Euler's totient function, which gives the number of integers ≤ m that are coprime to m.
    </p>
        <p>
      Some values for the length of F<sub>n</sub> given n:
      </p>
        <div class="informaltable">
          <table border="1">
            <colgroup>
              <col />
              <col />
            </colgroup>
            <thead>
              <tr>
                <th>n</th>
                <th>F<sub>n</sub></th>
              </tr>
            </thead>
            <tbody>
              <tr>
                <td>1</td>
                <td>2</td>
              </tr>
              <tr>
                <td>2</td>
                <td>3</td>
              </tr>
              <tr>
                <td>3</td>
                <td>5</td>
              </tr>
              <tr>
                <td>4</td>
                <td>7</td>
              </tr>
              <tr>
                <td>5</td>
                <td>11</td>
              </tr>
              <tr>
                <td>6</td>
                <td>13</td>
              </tr>
              <tr>
                <td>7</td>
                <td>19</td>
              </tr>
              <tr>
                <td>8</td>
                <td>23</td>
              </tr>
              <tr>
                <td>9</td>
                <td>29</td>
              </tr>
              <tr>
                <td>10</td>
                <td>33</td>
              </tr>
              <tr>
                <td>11</td>
                <td>43</td>
              </tr>
              <tr>
                <td>12</td>
                <td>47</td>
              </tr>
              <tr>
                <td>13</td>
                <td>59</td>
              </tr>
              <tr>
                <td>14</td>
                <td>65</td>
              </tr>
              <tr>
                <td>15</td>
                <td>73</td>
              </tr>
              <tr>
                <td>16</td>
                <td>81</td>
              </tr>
              <tr>
                <td>17</td>
                <td>97</td>
              </tr>
              <tr>
                <td>18</td>
                <td>103</td>
              </tr>
              <tr>
                <td>19</td>
                <td>121</td>
              </tr>
              <tr>
                <td>20</td>
                <td>129</td>
              </tr>
            </tbody>
          </table>
        </div>
        <p>
    </p>
      </div>
      <div class="refsect1">
        <a id="idp216345904"></a>
        <h2>Syntax</h2>
        <pre class="synopsis"><span class="command"><strong>f</strong></span> # time size "farey" fareynum mode</pre>
      </div>
      <div class="refsect1">
        <a id="idp216347936"></a>
        <h2>Initialization</h2>
        <p>
      <span class="emphasis"><em>size</em></span> -- number of points in the table. Must
      be a power of 2 or power-of-2 plus 1 (see <a class="link" href="f.html" title="f Statement (or Function Table Statement)"><em class="citetitle">f statement</em></a>).
    </p>
        <p>
      <span class="emphasis"><em>fareynum</em></span> -- the integer n for generating Farey Sequence F<sub>n</sub>
   </p>
        <p>
      <span class="emphasis"><em>mode</em></span> -- integer to trigger a specific
      output to be written into the table:
      </p>
        <div class="itemizedlist">
          <ul class="itemizedlist" style="list-style-type: disc; ">
            <li class="listitem">
              <p>0 -- outputs floating point numbers representing the
          elements of F<sub>n</sub>.</p>
            </li>
            <li class="listitem">
              <p> 1 -- outputs delta values of successive elements of
          F<sub>n</sub>, useful for generating note durations for
          example.</p>
            </li>
            <li class="listitem">
              <p> 2 -- outputs only the denominators of the integer
          ratios, useful for indexing other tables or instruments for
          example.</p>
            </li>
            <li class="listitem">
              <p> 3 -- same as mode 2 but with normalised
          output.</p>
            </li>
            <li class="listitem">
              <p> 4 -- same as mode 0 but with 1 added to each number,
          useful for generating tables for tuning opcodes, for example
          <span class="emphasis"><em>cps2pch</em></span>.
          </p>
            </li>
          </ul>
        </div>
        <p>
   </p>
      </div>
      <div class="refsect1">
        <a id="idp216358800"></a>
        <h2>Examples</h2>
        <p>
      </p>
        <div class="informalexample"><pre class="programlisting">
<span class="stamnt">f</span>1 0	-23 "farey" 8 0</pre>
        Generates generates Farey Sequence F<sub>8</sub>. The table contains all
        23 elements of F<sub>8</sub> as floating point numbers.
        
        <pre class="programlisting">
<span class="stamnt">f</span>1 0 -18 "farey" 7 1</pre>
This generates Farey Sequence F<sub>7</sub>. The table contains 18 delta values of F<sub>7</sub>,
i.e. the difference between r<sub>i+1</sub> - r<sub>i</sub>, where r is the ith element of F<sub>n</sub>.
 
        <pre class="programlisting">
<span class="stamnt">f</span>1 0	-43 "farey" 11 2</pre>
This generates Farey Sequence F<sub>11</sub>. The table contains the denominators of all
43 fractions in F<sub>11</sub>.

        <pre class="programlisting">
<span class="stamnt">f</span>1 0	-43 "farey" 11 3</pre>
This generates Farey Sequence F<sub>11</sub>. The table contains the denominators of all
43 fractions in F<sub>11</sub>, each of those divided by 11, i.e. normalised.

        <pre class="programlisting">
<span class="stamnt">f</span>1 0	-18 "farey" 7 4</pre>
This generates Farey Sequence F<sub>7</sub>. The table contains all fractions of F<sub>7</sub>,
same as mode 0, but this time '1' is added to each table element. 
      </div>
        <p>
    </p>
        <p>
      </p>
        <div class="example">
          <a id="idp216373280"></a>
          <p class="title">
            <strong>Example 1130. A simple example of the GENfarey routine.</strong>
          </p>
          <div class="example-contents">
            <p>
          See the sections <a class="link" href="UsingRealTime.html" title="Real-Time Audio"><em class="citetitle">Real-time Audio</em></a> and <a class="link" href="CommandFlags.html" title="Csound command line"><em class="citetitle">Command Line Flags</em></a> for more information on using command line flags.
        </p>
            <pre class="programlisting">
<span class="csdtag">&lt;CsoundSynthesizer&gt;</span>
<span class="csdtag">&lt;CsOptions&gt;</span>

<span class="csdtag">&lt;/CsOptions&gt;</span>
<span class="csdtag">&lt;CsInstruments&gt;</span>

<span class="ohdr">sr</span><span class="op">=</span>44100
<span class="ohdr">ksmps</span><span class="op">=</span>10
<span class="ohdr">nchnls</span><span class="op">=</span>1

<span class="oblock">instr</span> 4
      kndx <span class="opc">init</span> 0 <span class="comment">; read out elements of F_8 one by one and print to file</span>
      <span class="octrl">if</span> (kndx <span class="op">&lt;</span> 23) then    
      	 kelem <span class="opc">tab</span> kndx, 1
      	 <span class="opc">fprintks</span> "farey8table.txt", "<span class="op">%</span>2.6f\\n", kelem
      	 kndx <span class="op">=</span> kndx<span class="op">+</span>1
      <span class="octrl">endif</span>
<span class="oblock">endin</span>
<span class="csdtag">&lt;/CsInstruments&gt;</span>
<span class="csdtag">&lt;CsScore&gt;</span>
<span class="comment">; initialise integer for Farey Sequence F_8</span>
<span class="stamnt">f</span>1 0 -23 "<span class="stamnt">f</span>arey" 8 0
      <span class="comment">; if mode=0 then the table stores all elements of the Farey Sequence</span>
      <span class="comment">; as fractions in the range [0,1]</span>
<span class="stamnt">i</span>4	0     1
<span class="stamnt">e</span>
<span class="csdtag">&lt;/CsScore&gt;</span>
<span class="csdtag">&lt;/CsoundSynthesizer&gt;</span>
</pre>
          </div>
        </div>
        <p><br class="example-break" />
    </p>
        <p>
      Here is a complete example of the GENfarey routine. It uses the files <a class="ulink" href="examples/genfarey-2.csd" target="_top"><em class="citetitle">genfarey-2.csd</em></a>.
      </p>
        <div class="example">
          <a id="idp216378624"></a>
          <p class="title">
            <strong>Example 1131. Another example of the GENfarey routine.</strong>
          </p>
          <div class="example-contents">
            <p>See the sections <a class="link" href="UsingRealTime.html" title="Real-Time Audio"><em class="citetitle">Real-time Audio</em></a> and <a class="link" href="CommandFlags.html" title="Csound command line"><em class="citetitle">Command Line Flags</em></a> for more information on using command line flags.</p>
            <pre class="programlisting">
<span class="csdtag">&lt;CsoundSynthesizer&gt;</span>
<span class="csdtag">&lt;CsOptions&gt;</span>
<span class="comment">; Select audio/midi flags here according to platform</span>
-odac     <span class="comment">;;;realtime audio out</span>
<span class="comment">;-iadc    ;;;uncomment -iadc for RT audio input as well </span>
<span class="comment">; For Non-realtime ouput leave only the line below:</span>
<span class="comment">; -o genfarey.wav -W ;;; for file output any platform</span>
<span class="csdtag">&lt;/CsOptions&gt;</span>
<span class="csdtag">&lt;CsInstruments&gt;</span>
<span class="ohdr">sr</span> <span class="op">=</span> 44100
<span class="ohdr">ksmps</span> <span class="op">=</span> 32
<span class="ohdr">nchnls</span> <span class="op">=</span> 2
<span class="ohdr">0dbfs</span>  <span class="op">=</span> 1

<span class="comment">; GENfarey creates table gidelta. </span>
<span class="comment">; The table contains the delta values of Farey Sequence 7 (p5=7).</span>
<span class="comment">; They are used as Inter Onset Intervals (IOIs) or event durations.</span>
<span class="comment">; If p6 is set to 1 for IOI output then the length of the table (p3=-18) is -(|F_7| - 1)</span>
<span class="comment">; Remember that a negative sign is for non-power-of-2 table lengths.</span>
<span class="comment">; The negative sign in front of the GEN number prevents post-normalisation of its values.</span>

gidelta <span class="ohdr">ftgen</span> 0,0,<span class="op">-</span>18,"farey",7,1

<span class="comment">; Use GENfarey with p6 set to 2 to generate the denominators of fractions of F_7 </span>
<span class="comment">; this is used in this example as factors to create a series of pitches:</span>
gimult <span class="ohdr">ftgen</span> 0,0,<span class="op">-</span>18,"farey",7,2

<span class="comment">;-------- loop and trigger instrument 901 using a Farey Sequence polyrhythm</span>
	  <span class="oblock">instr</span> 1
kindx <span class="opc">init</span> 0
kindx2 <span class="opc">init</span> 0
ktrigger <span class="opc">init</span> 0
ktime_unit <span class="opc">init</span> p6
kstart <span class="opc">init</span> p4
kloop <span class="opc">init</span> p5
kinitndx <span class="opc">init</span> 0
kfn_times <span class="opc">init</span> gidelta
knote <span class="opc">init</span> 60
kbasenote <span class="opc">init</span> p8
ifundam <span class="opc">init</span> p7
ktrigger <span class="opc">seqtime</span> ktime_unit, kstart, kloop, kinitndx, kfn_times
  <span class="octrl">if</span> (ktrigger <span class="op">&gt;</span> 0 ) then
     kpitch <span class="op">=</span> <span class="opc">cpspch</span>(ifundam)
     kmult <span class="opc">tab</span> kindx2, gimult
     kpitch <span class="op">=</span> kpitch <span class="op">*</span> kmult
     knote <span class="op">=</span> kbasenote <span class="op">+</span> kmult
     <span class="opc">event</span> "i", 901,   0,   .4, .10, kpitch, kpitch <span class="op">*</span> .9, 0.4,  5,   .75, .8,  1.0, .15, .0,  .125, .125, .25, .5,  1.0, .0, .0,  .0,  .0,  .125, .25, .25, .25, knote
     kindx <span class="op">=</span> kindx <span class="op">+</span> 1
     kindx <span class="op">=</span> kindx <span class="op">%</span> kloop
     kindx2 <span class="op">=</span> kindx2 <span class="op">+</span> 1
     kindx2 <span class="op">=</span> kindx2 <span class="op">%</span> kloop
  <span class="octrl">endif</span>
<span class="oblock">endin</span>

<span class="comment">;------ basic 2 Operators FM algorithm ----------------</span>
	<span class="oblock">instr</span> 901
inotedur	<span class="op">=</span>		p3
imaxamp		<span class="op">=</span>		p4 <span class="comment">;ampdb(p4)</span>
icarrfreq	<span class="op">=</span>		p5
imodfreq	<span class="op">=</span>		p6
ilowndx		<span class="op">=</span>		p7
indxdiff	<span class="op">=</span>		p8<span class="op">-</span>p7
knote	        <span class="op">=</span>		p27
aampenv		<span class="opc">linseg</span>	p9, p14<span class="op">*</span>p3, p10, p15<span class="op">*</span>p3, p11, p16<span class="op">*</span>p3, p12, p17<span class="op">*</span>p3, p13 
adevenv		<span class="opc">linseg</span>	p18, p23<span class="op">*</span>p3, p19, p24<span class="op">*</span>p3, p20, p25<span class="op">*</span>p3, p21, p26<span class="op">*</span>p3, p22
amodosc		<span class="opc">oscili</span>	(ilowndx<span class="op">+</span>indxdiff<span class="op">*</span>adevenv)<span class="op">*</span>imodfreq, imodfreq, 10 
acarosc		<span class="opc">oscili</span>	imaxamp<span class="op">*</span>aampenv, icarrfreq<span class="op">+</span>amodosc, 10 
		<span class="opc">outs</span>		acarosc, acarosc  
<span class="oblock">endin</span>
<span class="csdtag">&lt;/CsInstruments&gt;</span>
<span class="csdtag">&lt;CsScore&gt;</span>
<span class="stamnt">f</span>10 0 4096 10 1	<span class="comment">;sine wave			</span>
<span class="comment">; p4 kstart  := index offset into the Farey Sequence</span>
<span class="comment">; p5 kloop   := end index into Farey Seq.</span>
<span class="comment">; p6 timefac := time in seconds for one loop to complete</span>
<span class="comment">; p7 fundam  := fundamental of the FM instrument</span>
<span class="comment">; p8 basenote:= root pitch of the midi voice output</span>
<span class="comment">; note that pitch structures of the midi file output are not equivalent to the</span>
<span class="comment">; ones used for the FM real-time synthesis.</span>

<span class="comment">;	start		dur		kstart	kloop   timefac	fundam. basenote</span>
<span class="stamnt">i</span>1	0.0		44		0 	18	2	6.05	60
<span class="stamnt">i</span>1	4		30		0 	18	3	7.05	72
<span class="stamnt">i</span>1	34		12		9 	18	3	7.05	72
<span class="stamnt">i</span>1	10		12		0 	18	1.5	8	84
<span class="stamnt">i</span>1	22		12		0 	9	1.5	8	84
<span class="stamnt">i</span>1	15		16		0	18	1	5	48
<span class="stamnt">i</span>1	22		20		5	17	1.7	4	36

<span class="stamnt">i</span>1	46		20		3 	11	2.5	7.04	71
<span class="stamnt">i</span>1	51		20		5 	13	2.5	7.06	72

<span class="stamnt">i</span>1	73.5		1.5		11	18	1.5	5.05	48
<span class="stamnt">i</span>1	75		1		12	18	1	6.03	58	
<span class="stamnt">e</span>
<span class="csdtag">&lt;/CsScore&gt;</span>
<span class="csdtag">&lt;/CsoundSynthesizer&gt;</span>
</pre>
          </div>
        </div>
        <p><br class="example-break" />

        <span class="phrase">These are the diagrams of the waveforms of the GENfarey routines, as used in the example:</span>

      </p>
        <div class="mediaobject">
          <img src="images/genfarey-2_1.png" alt="gidelta ftgen 100,0,-18,&quot;farey&quot;,7,1 - delta values of Farey Sequence 7" />
          <div class="caption">
            <p>gidelta ftgen 100,0,-18,"farey",7,1 - delta values of Farey Sequence 7</p>
          </div>
        </div>
        <p>

      </p>
        <div class="mediaobject">
          <img src="images/genfarey-2_2.png" alt="gimult ftgen 101,0,-18,&quot;farey&quot;,7,2 - generate the denominators of fractions of F_7" />
          <div class="caption">
            <p>gimult ftgen 101,0,-18,"farey",7,2 - generate the denominators of fractions of F_7 </p>
          </div>
        </div>
        <p>

    </p>
      </div>
      <div class="refsect1">
        <a id="idp216390144"></a>
        <h2>Credits</h2>
        <p>
      </p>
        <table border="0" summary="Simple list" class="simplelist">
          <tr>
            <td>Author: Georg Boenn</td>
          </tr>
          <tr>
            <td>University of Glamorgan</td>
          </tr>
          <tr>
            <td>2010</td>
          </tr>
        </table>
        <p>
    </p>
        <p>New in Csound version 5.13</p>
      </div>
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